Did Christ know who He was as a youth?

Scripture clearly states, that "Jesus Christ the same yesterday, and to day, and for ever" (Heb. 13:8). This means that if He in the age 30 new that He is Christ, then in His infancy too He knew that. This view is supported by 4-5 century Fathers. The places of the New Testament that seem to imply the contrary (for instance, His words about not knowing the day of Doom) are interpreted by Fathers of the Church as follows: any of human qualities that imply limitation, like hunger, fear, tears, ignorance, He accepted voluntarily and only when He wished that, and not involuntarily like us.

One of the best Biblical Scholars on the gospels that I know of implies that Christ was in many ways like a normal baby. Not only would he have not known that he was the Messiah when he was two, but he may not have been potty trained yet.

As far as I know his visit to the temple is seen as a potential place Jesus began to really become conscious that he was the Christ:

And yet this awakening of the Christ-consciousness on His first visit to the Temple, partial, and perhaps even temporary, as it may have been, seems itself like the morning-dawn, which from the pinnacle of the Temple the Priest watched, ere he summoned his waiting brethren beneath to offer the early sacrifice. (Alfred Edersheim, The Life and Times of Jesus the Messiah, P243)

This seems to be a valid suggestion because in many many places in the Bible even as an adult, where Jesus as a man was not aware of things that he naturally knew as God.

Rather:

And Jesus grew in wisdom and stature, and in favor with God and man. (NIV Luke 2:52)

1. As has been stated in one of the other answers the Lord Jesus made at the age of twelve a public distinction between His Father and Joseph and stating that He must be about His Father's business (KJV); not house (ESV). Else He would have spend the rest of His earthly days in the Temple...

Luke 2:41-52 (KJV)

Now his parents went to Jerusalem every year at the feast of the passover. And when he was twelve years old, they went up to Jerusalem after the custom of the feast. And when they had fulfilled the days, as they returned, the child Jesus tarried behind in Jerusalem; and Joseph and his mother knew not of it. But they, supposing him to have been in the company, went a day's journey; and they sought him among their kinsfolk and acquaintance. And when they found him not, they turned back again to Jerusalem, seeking him. And it came to pass, that after three days they found him in the temple, sitting in the midst of the doctors, both hearing them, and asking them questions. And all that heard him were astonished at his understanding and answers. And when they saw him, they were amazed:and his mother said unto him, Son, why hast thou thus dealt with us? behold, thy father and I have sought thee sorrowing. And he said unto them, How is it that ye sought me? wist ye not that I must be about my Father's business? And they understood not the saying which he spake unto them. And he went down with them, and came to Nazareth, and was subject unto them:but his mother kept all these sayings in her heart. And Jesus increased in wisdom and stature, and in favour with God and man

2. He also throughout his ministry never called Mary "Mother" but rather "Woman". Respectfully, but to not place any spiritual authority with her as Roman Catholicism does.

John 2:3-5

And when they wanted wine, the mother of Jesus saith unto him, They have no wine. Jesus saith unto her, Woman, what have I to do with thee? mine hour is not yet come. His mother saith unto the servants, Whatsoever he saith unto you, do it.

John 19:25-27

Now there stood by the cross of Jesus his mother, and his mother's sister, Mary the wife of Cleophas, and Mary Magdalene. When Jesus therefore saw his mother, and the disciple standing by, whom he loved, he saith unto his mother, Woman, behold thy son! Then saith he to the disciple, Behold thy mother! And from that hour that disciple took her unto his own home.

3. For the Lord Jesus Christ to be a sinless sacrifice must mean that He never sinned, not even as a child. For a child not to sin takes some extraordinary, supernatural work because it is simply an impossibility. Every child sins in so many ways even when they are but a few weeks/months old; as any parent can testify when watching that first tantrum... So for the Lord not to have sinned must mean that He not only was aware of His sinless holy Nature, but that He was by this Godly Nature able to restrain Himself from sinning, even as a little babe.

Hebrews 4:15

For we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin.

We know almost nothing for sure about the life of Jesus before starting his public ministry. From the Gospels we have only the birth narratives and "boy Jesus in the temple", which are really very little to go on. From those we can deduce a couple of things:

• Mary knew quite a lot about the destiny of her son, that he would "redeem his people". It's not known whether she chose to pass this knowledge on to Jesus. I don't know if that counts as 'knowing' in the way you mean.
• Jesus clearly knew he was unusual in some way, from the temple narrative. He must have known that it was not usual for young boys to be disputing theology with the teachers; at least he would have been told that afterwards
• Jesus clearly had foreknowledge of his destiny by the start of his public ministry

There are a number of ancient extra-biblical traditions regarding the infant Jesus, in which he performs miracles. The Infancy Gospel of Thomas is an example. If these have any credence at all then they would tend to support the view that Jesus at least knew he was not a normal child at an early age.

Luke 2 tells the story of Jesus as a boy in the Temple. His parents had taken Him to a feast there and had left Him behind on their return trip. When they returned to Jerusalem to find Him, Jesus asks His mother a question--"Did you not know that I must be in my Father's house?"

Now, the Temple was not where Joseph lived, but was the House of God. So, from that very early age, Jesus understood who He was and that His Father was not Joseph, but God the Father.

Now his parents went to Jerusalem every year at the Feast of the Passover. And when he was twelve years old, they went up according to custom. And when the feast was ended, as they were returning, the boy Jesus stayed behind in Jerusalem. His parents did not know it, but supposing him to be in the group they went a day's journey, but then they began to search for him among their relatives and acquaintances, and when they did not find him, they returned to Jerusalem, searching for him. After three days they found him in the temple, sitting among the teachers, listening to them and asking them questions. And all who heard him were amazed at his understanding and his answers. And when his parents saw him, they were astonished. And his mother said to him, “Son, why have you treated us so? Behold, your father and I have been searching for you in great distress.” And he said to them, “Why were you looking for me? Did you not know that I must be in my Father's house?” And they did not understand the saying that he spoke to them. And he went down with them and came to Nazareth and was submissive to them. And his mother treasured up all these things in her heart. Luke 2:41-51 ESV